This section is intended to introduce the reader to various aspects of art, which may be related to various aspects of the present invention that are described and/or claimed below. This discussion is believed to be helpful in providing the reader with background information to facilitate a better understanding of the various aspects of the present invention. Accordingly, it should be understood that these statements are to be read in this light, and not as admissions of prior art.
A pixel beam (as depicted in FIG. 1, and referenced 10) represents a volume occupied by a set of rays of light in an object space of an optical system referenced 11 of an optical device (not shown on FIG. 1, that can be either a conventional camera or a light field acquisition camera). Hence, a given pixel referenced 12 of an image sensor referenced 13, is associated with a pixel beam (i.e. a set of rays of light in the object space) that is detected by said given pixel 12, and that passes through a pupil 14 of said optical system 14. The optical system 14 may be a combination of lenses fit for photo or video cameras. A pupil of an optical system is defined as the image of an aperture stop as seen through said optical system, i.e. the lenses of the optical acquisition system, which precedes said aperture stop. An aperture stop is an opening which limits the amount of light which passes through the optical system of the optical acquisition system. For example, an adjustable blade diaphragm located inside a camera lens is the aperture stop for the lens. The amount of light admitted through the diaphragm is controlled by the diameter of the diaphragm opening which may adapt depending of the amount of light a user of the camera wishes to admit. For example, making the aperture smaller reduces the amount of light admitted through the diaphragm, and, simultaneously, increases the depth of focus. The apparent size of a stop may be larger or smaller than its physical size because of the refractive action of a portion of the lens. Formally, a pupil is the image of the aperture stop through all lenses of the optical acquisition system located between the physical stop and the observation space.
More precisely, a pixel beam 10 can be defined as a pencil of rays of light that reach a given pixel 22 when propagating through the optical system 11 via an entrance pupil 14. As light travels on straight lines in free space, the shape of such a pixel beam 10 can be defined by two sections, one being the conjugate 15 of the pixel 12, and the other being the entrance pupil 14. The pixel 12 is defined by its non-null surface and its sensitivity map.
Thus, a pixel beam also referenced 20, as shown on FIG. 2, may be represented by a hyperboloid of one sheet supported by two elements: the pupil referenced 24 and the conjugate referenced 25 of the pixel 12 in the object space.
A hyperboloid of one sheet is a ruled surface that can support the notion of pencil of rays of light and is compatible with the notion of “étendue” of physical light beams, notion linked to the preservation of energy across sections of the physical light beams.
As represented on FIG. 3, a hyperboloid of one sheet referenced 30 is mostly identical to its asymptotic cones referenced 31, 32, except in the fundamental region of its smallest section, called the waist referenced 35, which corresponds to the conjugate 15 in the object space. For plenoptic systems, such as light-field cameras, this is the region where space sampling by multiple path rays is performed. Sampling space with cones in this region is not adequate, as pixel 12 sensitivity is significant on some tens of square microns on its surface and cannot be represented by a mathematical point with infinitely small surface as would be a cone tip.
It is possible to define each pixel beam 10, 20, 30 by four independent parameters: zP,θx,θy, a defining the position and size of the pixel conjugate 15, 35, in front of the pupil 14, 24 and by six pupilar parameters x0,y0,z0,θx0,θy0, r which define the position, orientation and radius of the pupil 14, 24. These six pupilar parameters are common to the collection of pixel beams sharing a same pupil 14, 24. Indeed, a pixel beam represents the volume occupied by a set of rays of light in the object space of the optical system 11 sensed by the pixel 12 through the pupil 14, i.e. to a given couple pixel 12/pupil 14, 24 corresponds a unique pixel beam 10, 20, 30, but a plurality of distinct pixel beams can be supported by a same pupil 14, 24.
An origin O of a coordinate system (x,y,z) in which the parameters of the pixel beam 10, 20, 30 are defined corresponds to the center of the pupil 14 as shown on FIG. 1, where the z axis defines a direction normal to the surface of the pupil 14, 24.
Usually, as represented on FIG. 4, the first ray to be considered for describing a pixel beam is its axis or chief ray referenced 41. The chief ray 41 corresponds to the z axis of the hyperboloid 10, 20, 30, as represented on FIG. 2.
Indeed, the parameters θx,θy, define chief ray directions relative to the entrance of the pupil 14 center. They depend on the pixel 12 position on the sensor 13 and on the optical elements of the optical system 11. More precisely, the parameters θx,θy represent shear angles defining a direction of the conjugate 15 of the pixel 12 from the center of the pupil 14.
The parameter zP represents a distance of the waist 35 of the pixel beam 10, 20, 30, or the conjugate 15 of the pixel 12, along the z axis.
The parameter a represents the radius of the waist 35 of the pixel beam 10, 20, 30.
For optical systems 11 where optical distortions and field curvatures may be modelled, the parameters zP and a can depend on the parameters θx and θy via parametric functions.
The four independent parameters are related to the pixel 12 and its conjugate 15.
The six complementary pupilar parameters defining a pixel beam 10, 20, 30 are:                r which represents the pupil 14, 24 radius,        x0,y0,z0 which represent the coordinates of the pupil 24, 34 center in the (x,y,z) coordinate system, and        θx0,θy0 which represent the orientation of the pupil 14, 24 in the reference (x,y,z) coordinate system.        
These six pupilar parameters are related to the pupil 14, 24. Another parameter c can be defined. Such a parameter c is dependent on the parameters zP and a related to the pixel 12 and its conjugate 15 and on the parameters r related to the pupil 14, 24. The parameter c defines the angular aperture a of the pixel beam 10, 20, 30 and is given by the formula
      tan    ⁡          (      α      )        =            a      c        .  
Thus, the expression of the parameter c is given by the following equation:
                              c          2                =                                            a              2                        ⁢                          z              P                                                                                ⁢                2                                                                        r              2                        -                          a                                                                                ⁢                2                                                                        (        1        )            
The coordinates (x,y,z), in the object space, of points belonging to the surface delimiting the pixel beam 10, 20, 30 are function of the above defined sets of parameters related to the pupil 14, and to the conjugate 15 of the pixel. Thus, equation (2) enabling the generation of the hyperboloid of one sheet representing the pixel beam 10, 20, 30 is:
                                                                        (                                  x                  -                                      z                    ·                                          tan                      ⁡                                              (                                                  θ                          x                                                )                                                                                            )                            2                                      a                                                                                ⁢                2                                              +                                                    (                                  y                  -                                      z                    ·                                          tan                      ⁡                                              (                                                  θ                          y                                                )                                                                                            )                            2                                      a                                                                                ⁢                2                                              -                                                    (                                  z                  -                                      z                    P                                                  )                            2                                      c              2                                      =        1                            (        2        )            
Hence, there is a need to provide a technique for determining (via an approximation or an estimation) these parameters that defines a pixel beam associated with a pixel on an image sensor via the execution of a calibration process for a given optical device.